| L(s) = 1 | + 2-s + 1.82·3-s + 4-s + 5-s + 1.82·6-s − 4.26·7-s + 8-s + 0.340·9-s + 10-s + 3.40·11-s + 1.82·12-s − 2.12·13-s − 4.26·14-s + 1.82·15-s + 16-s − 4.20·17-s + 0.340·18-s + 3.30·19-s + 20-s − 7.80·21-s + 3.40·22-s + 4.52·23-s + 1.82·24-s + 25-s − 2.12·26-s − 4.86·27-s − 4.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s + 0.447·5-s + 0.746·6-s − 1.61·7-s + 0.353·8-s + 0.113·9-s + 0.316·10-s + 1.02·11-s + 0.527·12-s − 0.589·13-s − 1.14·14-s + 0.471·15-s + 0.250·16-s − 1.01·17-s + 0.0802·18-s + 0.758·19-s + 0.223·20-s − 1.70·21-s + 0.726·22-s + 0.943·23-s + 0.373·24-s + 0.200·25-s − 0.417·26-s − 0.935·27-s − 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.511763044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.511763044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 + 0.623T + 71T^{2} \) |
| 73 | \( 1 + 6.78T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 0.945T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61381692688604295273509212118, −6.90343420798042818727288194995, −6.52664744087135571611223013155, −5.80158543601734418269227125929, −4.94940258001927289417488262118, −3.99288096755942123520778271925, −3.46369808432445227198612844935, −2.69636766949005910467975852612, −2.28857043771429622582480773412, −0.873179924323584850419589983346,
0.873179924323584850419589983346, 2.28857043771429622582480773412, 2.69636766949005910467975852612, 3.46369808432445227198612844935, 3.99288096755942123520778271925, 4.94940258001927289417488262118, 5.80158543601734418269227125929, 6.52664744087135571611223013155, 6.90343420798042818727288194995, 7.61381692688604295273509212118
